Orientation averaging of electron backscattered diffraction data


Professor F. J. Humphreys. Fax: +44 (0)161 200 8877; e-mail: john.humphreys@umist.ac.uk


The use of data averaging to improve the angular precision of electron backscattered diffraction (EBSD) maps is discussed. It is shown that orientations may be conveniently and rapidly averaged using the four Euler-symmetric parameters which are coefficients of a quaternion representation. The processing of EBSD data requires the use of an edge preserving filter and a modified Kuwahara filter has been successfully implemented and tested. Three passes of such a filter have been shown to reduce orientation noise by a factor of ∼10. Application of the method to deformed and recovered aluminium alloys has shown that such data processing enables small subgrain misorientation (< 0.5°) to be detected reliably.

1. Introduction

Electron backscattered diffraction (EBSD) in the scanning electron microscope (SEM) is now well established as a powerful analytical tool which is being routinely used by metallurgists (Wilkinson & Hirsch, 1997; Humphreys, 1999a), ceramicists (Saylor & Rohrer, 1999) and geologists (Prior et al., 1999) to investigate microstructure and crystallographic texture. The technique is based on the use of diffraction patterns from bulk samples in the SEM, and has been developed over a number of years (see e.g. Randle, 1992). The most important recent advance has been the development of rapid automated diffraction pattern analysis, which, together with control of the microscope beam or stage, has enabled line or area scans (orientation maps) of a sample surface to be obtained rapidly and automatically (Wright & Adams, 1992) as shown in Fig. 1. EBSD is increasingly being used as a method for characterizing microstructures and textures and is becoming a routine tool for quantifying the parameters relating to grains, subgrains and multiphase microstructures (Humphreys, 1999a). The use of a field emission gun (FEGSEM) for EBSD has improved the spatial resolution to the extent that EBSD can now be used for many of the investigations which have traditionally been carried out by transmission electron microscopy (TEM) (Humphreys & Brough, 1999; Humphreys, 1999b). Specific advantages of EBSD over TEM for microstructural characterization are that large solid samples may be used and that complete orientation data are available for each point (pixel) in the map.

Figure 1.

EBSD map of aluminium alloy with a grain size of ∼0.5 µm.

Deformed or recovered microstructures typically contain cells or subgrains of size ∼0.2–1 µm and misorientation of ∼0.5–3° (e.g. Humphreys & Hatherly, 1995) and characterization of the size and orientation distributions of such microstructures requires a combination of good spatial and angular resolution at the operating conditions for which data may be acquired at a reasonable rate. For aluminium alloys the typical requirement for quantitative microstructural measurement would be for a map containing ∼300 grains or subgrains, with at least 5 pixels across a grain or subgrain.

Under conditions of rapid data acquisition, the spatial resolution for EBSD in a FEGSEM on aluminium is ∼20 × 60 nm (Humphreys & Brough, 1999) and therefore samples with grains or subgrains as small as ∼0.1 × 0.3 µm can be measured. The minimum map size for quantitative grain or subgrain measurement would therefore be ∼100 × 100 pixels. Current EBSD systems can acquire data at the rate of 5–10 pixels per second and therefore such a map could be acquired in ∼30 min.

The relative angular precision between adjacent pixels for a rapid data acquisition EBSD system is ∼1° (Humphreys & Brough, 1999) and this therefore becomes a limiting factor in examining deformed or recovered microstructures. The poor angular resolution also makes determination of the axis of misorientation between subgrains of low misorientation extremely difficult (Prior, 1999). It is expected that the manufacturers of commercially available EBSD systems will implement improvements in angular resolution in the future through more accurate acquisition and analysis of the diffraction patterns.

In a typical orientation map such as shown in Fig. 1, which contains 265 grains in a 100 × 100 pixel map, there are a considerable number of data points (an average of 38) within each grain, and the orientation of each grain could be more precisely defined if averaging of the data within a grain were to be carried out. The potential benefits of such a process can be determined from a simple statistical analysis. If the accuracy of a single orientation measurement is 1°, then the 95% confidence limit in the orientation of a grain will fall with the number (n) of pixels averaged, as n−1/2 as shown in Fig. 2.

Figure 2.

Predicted improvement in orientation precision from data averaging.

Orientation averaging is only valid if the true orientation is constant within each grain or subgrain, and this should be considered before such procedures are used. In annealed aluminium, such as the sample in Fig. 1, this is known to be a good assumption. However, some materials may contain many free dislocations and only poorly developed boundaries, in which case orientation averaging may not be appropriate. Wherever possible, the nature of the microstructure should be checked using other techniques such as TEM- or SEM-based channelling contrast imaging (Newbury et al., 1986; Prior et al., 1996, Wilkinson & Hirsch, 1997), before orientation averaging is used. This point is further considered in section 5.

2. Orientation averaging

It is not a trivial matter to average orientations because although only three independent angular parameters are required to define a rotation, and so an orientation, in three spatial dimensions, there is no set of such parameters which facilitates operations such as orientation averaging in a satisfactory way. Commonly used three-parameter descriptions of orientation include Euler angles and axis-angle, along with associated sets such as the Rodrigues vector. Spaces using these parameters as bases are very distorted, and generally not even closed, and distances between orientations in them can differ greatly from the misorientation angle, which can be taken as a true measure of the misorientation distance. The problems associated with the statistical analysis of orientation data and various possible methods of orientation averaging are reviewed and discussed by Krieger Lassen et al. (1994) and Humbert et al. (1996).

The use of four-parameter descriptors of orientation appears to be the best solution, particularly if we are only interested in averaging orientations spread over a range of a few degrees. The Euler-symmetric parameters are given by:

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where d is the axis with a common numerical description in both the crystal and reference axes and φ is the rotation angle about that axis that completes the orientation specification. The axis is a direction, with length unity. As a result, the Euler-symmetric parameters can be thought of as a vector (x = {ρ, λ, µ, ν} will be used here) with unit length in four dimensions, and all orientations lie on the surface of a four-dimensional hypersphere with unit radius (a ‘hemi-hypersphere’ will do, as x and –x represent the same orientation). This is analogous to the way that directions in three dimensions, actually requiring only two parameters for a full specification, are conveniently represented as points on the surface of a three-dimensional sphere (although then usually stereographically projected to a plane), or even the way that a single angular variable is represented in two dimensions on compass dials, for example.

Potential benefits of using the Euler-symmetric parameters − often referred to as the quaternion description, although they are only coefficients of a quaternion − in crystallographic texture have been recognized by several workers, for example Frank (1988) and Ibe (1994). Their single most important property, for present purposes at least, is that the scalar product of two x vectors is the cosine of half the misorientation angle, ω, between the orientations they represent:

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Now, a conventional average (arithmetic mean) is the value that has the minimum sum of squared distances to the values in a data set. It makes little difference, especially with reasonably restricted ranges, to replace ‘minimizing the sum of squared distances’ with ‘maximizing the sum of cosines of half the distances’. That is because:

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An average value for orientation, , is then given by:

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which, as the scalar product is distributive, is equivalent to:

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Because all x vectors are of unit length, this means that a value for average orientation is given by adding the x vectors of those orientations, and normalizing the result to give the unit length. This simple technique has, of course, been known for some time (see Krieger Lassen et al., 1994) and Humbert et al. (1996) have discussed its application to the determination of mean orientations. A schematic showing the procedure reduced to two dimensions is given in Fig. 3.

Figure 3.

Schematic showing, in two dimensions, the simple orientation averaging process of summing orientation vectors, x, and normalizing to give unit radius.

In practice, there are a few further issues. Data from EBSD are often affected by symmetry, which can make orientations that are close appear highly misoriented. Data to be averaged are, then, transformed by appropriate crystal point group rotations to give the smallest misorientations with respect to a reference orientation. The reference orientation in the case of smoothing an orientation map will be the value at the point currently being smoothed. A simple test in the case of cubic symmetry is to check if ω < 45°: in that case, no symmetric equivalents need be generated. It has proved useful in practice to exclude orientations from this local averaging which deviate significantly − by some set cut-off misorientation − from the reference orientation, after reducing the misorientation by symmetry operations if necessary.

3. Smoothing strategy

It is a simple matter to take a grid of measured orientations, and then obtain new orientation estimates by averaging over a set of values centred on each grid point in turn. A complete new set needs to be calculated, rather than sequentially replacing orientation values which can lead to cumulative, ‘marching’, errors. However, such a procedure tends to remove the features in the microstructure which are of greatest interest: grain and subgrain boundaries.

The preservation of edges whilst smoothing is a common problem in image processing, and a well known technique for dealing with it is to use strategies such as the Kuwahara filter (Kuwahara & Eiho, 1976). In this, the array of grid points surrounding the (central) point being smoothed is subdivided into regions, each including that central point. The sub-region with the least variance is used to calculate the mean value, which then replaces the centre point value after completion of the whole grid. With orientations, ‘least variance’ is replaced by ‘maximum sum of cos(ω/2) relative to the mean, divided by (N − 1)’. The calculation of this is straightforward using Euler-symmetric parameter vectors. The ‘inverse variance’ measure is simply the normalization factor involved in obtaining the mean, divided by (N − 1). That division can be dispensed with if all samples are the same size. It is clear from Fig. 3 that the length of a sum of a given number of x increases as they become more closely parallel.

The size and shape of both the whole sampling region and the sub-regions can have significant effects on the smoothing process. A very simple strategy is shown in Fig. 4 in which four sub-regions of 3 × 3 are tested within a 5 × 5 block. Increasing the number of the sub-regions, e.g. eight sub-regions of 3 × 3 in the 5 × 5 block, allows better edge preservation at corners, a point which is further considered in section 4.

Figure 4.

A simple Kuwahara smoothing strategy. A 5 × 5 region is used, light grey in (a), about the point to be smoothed (darker grey). The mean and ‘variance’ in each of four sub-regions, shown darker in (b–e), is calculated and the average value from the region with least ‘variance’ is assigned to the new orientation of the central point.

4. Implementation and testing

Routines for EBSD data smoothing were incorporated into a program (VMAP), which has been developed in-house for the analysis and presentation of EBSD map data. Routines for simple smoothing of a 3 × 3 data block and a Kuwahara filter which used eight blocks of 3 × 3 selected from a 5 × 5 matrix were implemented.

It is interesting to compare simple smoothing with that using the Kuwahara filter, and Fig. 5(a) shows the effect of smoothing on a set of data created by adding random orientation noise to a reference orientation. The noise, which gave a maximum deviation from the reference orientation of 10°, is very much greater than that obtained from EBSD. Both methods significantly reduce the orientation noise with one or two passes. Figure 5(b) shows the smoothing and filtering of data typical of EBSD (maximum deviation of 2° from the reference orientation. It is seen that two passes reduces the orientation noise by a factor of ∼10, which is a significant and worthwhile improvement. It is clear from Fig. 5a that simple smoothing reduces the noise more rapidly than the Kuwahara filter, and it is noticeable that the latter is ineffective after ∼ four passes. This is because the edge preservation routine tends to organize the data into blocks of similar orientation, as shown in Fig. 6, and once formed, these blocks are stable. The mean misorientation of these blocks is found to be approximately 0.1 of the original orientation noise. It must be emphasized that such domains are a result of the data processing and should not be interpreted as microstructural features.

Figure 5.

(a) Smoothing of very noisy data. (b) Smoothing with Kuwahara filter of data typical of EBSD.

Figure 6.

Smoothing with Kuwahara filter (six runs) showing blocky data. The misorientation between blocks is typically ∼0.1°.

In some cases the use of a standard Kuwahara filter may introduce small regions of spurious orientation into a map, as shown in Fig. 7, which is a schematic representation of a triple point in a map. The region contains three grains or subgrains whose orientations are denoted 1, 2 and 3. The standard Kuwahara filter will work correctly for most points, but not for the dark shaded pixel in grain 1. The 3 × 3 block with least variance is likely to be the lightly shaded block and the mean orientation calculated for the dark pixel will be incorrect because it averages over grains 1 and 2. This problem can be averted if the filter is modified so as to abort the averaging procedure for a particular point if the block with lowest variance contains a pixel misoriented to the reference pixel by more than a set minimum value. This value is set by the user, and we typically use 3°.

Figure 7.

The effect of filtering on grain boundary corners.

Figure 8 shows the effect of filtering on an artificial microstructure containing low angle boundaries. Figure 8a is a map of the idealized microstructure containing triangular grains misoriented from the background grain by the amounts indicated. Boundaries with misorientations larger than 0.5° are drawn in white. In Fig. 8b, orientation noise of a similar intensity to that of Fig. 5b is introduced and many additional boundary regions which are due to the noise are seen. Figure 8c shows the map after two passes of the modified Kuwahara filter. It is seen that the spurious boundaries have been removed and the original triangular grains are reasonably well preserved.

Figure 8.

Effect of filtering on low angle grain boundaries. (a) Artificial EBSD map containing small triangular grains misoriented by 0.4–1.2° from the matrix grain. 0.5° grain boundaries are shown as white lines. (b) Orientation noise at the same level as Fig. 5(b) introduced. (c) Effect of two passes of the modified Kuwahara filter.

5. Performance with real EBSD data

Extensive use has been made of the smoothing and filtering routines discussed above for the analysis of EBSD data, and in particular, we are using it to improve the angular resolution in the study of deformed and recovered aluminium alloys. Figure 9 shows an example of such an application. The sample is a high purity Al-0.1wt%Mg alloy which has been deformed 20% by cold rolling and annealed at 200 °C for 1 h. The sample was sectioned perpendicular to both the rolling direction (RD) and the normal direction (ND), and in the maps of Fig. 9, RD is vertical. TEM showed that the recovered microstructure consisted of subgrains, which were often rectangular in the section examined, and that there were few free dislocations within the subgrains.

Figure 9.

EBSD maps of the deformed and recovered aluminium alloy described in the text. (a–d) see facing page. (a) Pattern quality map. (b) Raw data. (c) Raw data with boundaries > 0.5° superimposed. (d) Data after two passes of modified Kuwahara filter. (e) Processed data with boundaries > 0.5° superimposed. The colours in (b)–(e) represent ‘relative Euler contrast’. The line in (b) shows the line of data used for Fig. 10.

EBSD was carried out on electropolished samples in a Camscan Maxim 2040SF FEGEM at 20 keV, using an HKL Channel acquisition system with a Nordif CCD camera. The data of Fig. 9 are part of a larger map with a pixel step size of 0.2 µm and are taken from the interior of a single grain.

Figure 9(a) is a ‘pattern quality’ map in which a measure of the contrast of the diffraction pattern corresponding to each pixel is converted to a grey level and plotted. Regions of perfect crystal give high quality patterns (light) and regions adjacent to boundaries give lower quality patterns (dark) because of overlapping diffraction lines. This type of map gives a good indication of the location of boundaries (and other large defects) in the sample, although there may be some ambiguity of the data, and such maps cannot readily be used for quantitative analysis.

Figure 9(b) shows the raw EBSD data. The colours used represent ‘relative Euler contrast’, which is defined in the Appendix. Figure 9(b) clearly shows the subgrains, but the data are noisy. If low angle grain boundaries of misorientations larger than 0.5° are superimposed on the map as in Fig. 9(c), then it is seen that many of the ‘boundaries’ lie inside the subgrains. Comparison with the boundaries revealed in the pattern quality map of Fig. 9(a), and TEM experiments, shows that many of the apparent boundaries are undoubtedly due to orientation noise. Figure 9(d) shows the effect of two passes of the modified Kuwahara filter on the raw data. The orientation noise is significantly reduced and the subgrains are more clearly visible. In Fig. 9(e), the low angle grain boundaries > 0.5° are superimposed on the map and in contrast to Fig. 9(c), the detected boundaries clearly match the microstructure.

The extent of the improvement in the data is also seen from a scan of the data along the line shown in Fig. 9(b), which is shown in Fig. 10. Figure 10(a) shows the misorientation angle of pixels in the raw data map, relative to the initial pixel in the scan, and Fig. 10(b) shows the same scan for the smoothed and filtered data of Fig. 9(d). Figure 10(b) clearly reveals the subgrains as regions of constant orientation, whereas analysis of Fig. 10(a) is more difficult.

Figure 10.

Scans of the data of Fig. 9 along the line shown in Fig. 9(b). The misorientation of a pixel is calculated relative to the initial point on the scan. (a) Raw data of Fig. 9(b). (b) Smoothed and filtered data of Fig. 9(d).

Comparison of Fig. 9(e) with the EBSD pattern quality map of Fig. 9(a) shows good correspondence of the boundaries, and this result when taken together with transmission microscopy of similar samples, confirms that the real microstructure is revealed in Fig. 9(e). The subgrain sizes, shapes and misorientations and their distributions may be retrieved rapidly and automatically from the processed data from maps such as Fig. 9.

6. Conclusions

1 Orientations may be conveniently and rapidly averaged using the four Euler-symmetric parameters which are coefficients of a quaternion representation.

2 The processing of EBSD data requires the use of an edge preserving filter and a modified Kuwahara filter has been successfully implemented and tested. Three passes of such a filter have been shown to reduce orientation noise by a factor of ∼10.

3 Application to deformed and recovered aluminium alloys has shown that such data processing enables small misorientations (< 0.5°) to be reliably detected.


The support of EPSRC and Alcan International for the work is gratefully acknowledged. Thanks are also due to Ian Brough for his help with the microscopes and EBSD equipment.


Appendix – relative Euler contrast

In plotting EBSD maps, there are a number of methods commonly used to allocate the colours or grey levels of the pixels and one of the most common is ‘Euler contrast’, in which the intensities of red, green and blue for the pixel are proportional to the values of the three Euler angles φ, φ1 and φ2 which define the orientation of the point. This is satisfactory for revealing the contrast between highly misoriented grains, but if the orientations within a region are similar, as would be the case for subgrains within a grain, there is little contrast between the subgrains. In such a situation we have found that suitable contrast can be produced using ‘relative Euler contrast’, which is used for the first time in this paper in Figs 6, 8 and 9.

A reference point in the map is selected, with Euler angles φ°, φ°1 and φ°2. For any pixel in the map with Euler angles φp, inline image, the intensities of red green and blue are then given by

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The parameter S, which is set by the user, determines the contrast stretch, and the addition of 128 ensures that positive and negative differences in Euler angles result in a valid positive intensity.

For example, the value of S used in Fig. 9 is 25°−1, which means that a colour does not saturate (255) or fall to zero provided that the magnitude of Δφ is less than 5°, i.e. this setting is optimum for a grain with a misorientation spread of ∼10°.